For fixed integers p, q an edge coloring of a complete graph K is called a (p, q)-coloring if the edges of every Kp ⊆ K are colored with at least q distinct colors. Clearly, (p, 2)-colorings are the classical Ramsey colorings without monochromatic Kp subgraphs. Let f(n,p,q) be the minimum number of colors needed for a (p,q)-coloring of Kn. We use the Local Lemma to give a general upper bound for f. We determine for every p the smallest q for which f(n,p,q) is linear in n and the smallest q for which f(n,p,q) is quadratic in n. We show that certain special cases of the problem closely relate to Turán type hypergraph problems introduced by Brown, Erdos and T. Sós. Other cases lead to problems concerning proper edge colorings of complete graphs.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics